Measurement: Making Conversions

I’ll admit, making conversions with measurement has always been difficult for me. Probably because I don’t apply this type of math on a daily basis (as you can most likely say for most people unless they do it regularly for their jobs). Add the fact that we teach the metric system, but don’t really use it. In researching a good way to teach measurement conversions, especially for 5th grade and up, I came upon a strategy which I will share below. If you try it, let me know how it works in your class.  I’m also going to share the visual for standard liquid measures as I believe it really helps think about how many cups in a quart, pints in a gallon, etc.

Here’s the liquid measure guide

G=gallon, Q= quart, P=pint, C=cup. An interesting tidbit regarding the words:  cup is the smallest unit and it has just 3 letters. Pint is next in size with 4 letters. Quart has 5 letters. Gallon has 6 letters.  So just thinking about the size of the word might be enough for some students to relate to these units.

Steps for students to make this:

1. Have 4 different markers ready, one for each unit.  I recommend students draw with pencil first, then trace with marker because they most likely will have to try more than once to make the shape a good size.
2. Make a giant capital G in one color. Try to make it take up almost the whole page with a vertical orientation. I tell them to square it off (like shown in the picture).
3. Then draw 4 Q’s inside as shown (different color). I kind of square them off as well to make room for the other parts. 4 quarts = 1 gallon.
4. Draw 2 P’s inside each Q (a third color). 2 pints = 1 quart
5. Draw 2 C’s inside each P (a fourth color).  2 cups = 1 pint
6. Now practice making equalities with various questions: How many cups in a quart? How many cups in a gallon? How many pints in 2 quarts, etc.

Other measurement conversions (metric, standard, length, liquid, etc.)

This method was described on the NCTM forum by a high school teacher, which I saved a few years ago. I hadn’t thought about it until recently when I needed to work with a 5th grader. I know there are rules out there like this:  Going from a smaller unit to a larger unit = divide; Going from a larger unit to a smaller unit = multiply.  But it’s always helpful to have 2 strategies. If you can’t remember whether to multiply or divide, then this strategy will help do it for you.

I think the illustrations speak for themselves, but the keys are as follows:

1. In step 1, rewrite the problem in fraction form. Place the labels of the units diagonally across from each other. This is so they will “cancel out”. Place an “x” sign. As you will notice in step 3, the way the fraction is written will determine whether you multiply or divide (which relates to the above . . . smaller to larger unit = divide.; larger to smaller unit = multiply).
2. In step 2, determine how many ___ in 1 _____. If needed, there are many charts available on TPT or Pinterest to help reference the correct conversion regarding customary or metric systems.
3. In step 3, complete the equation.

Have a terrific week! Happy measuring!

24 Summer Time Math Activities which can be done at home!

I realize many of you  (teachers and parents) may be searching for ways to link every day activities to math so that children can learn in a practical way while at home during this surrealistic period.  Happy Fourth of July and . . . .Here’s a list of things you might like to try:

Outdoors

1. While bouncing a ball, skip count by any number. See how high you get before missing the ball. Good to keep your multiplication facts current.
2. How high can you bounce a ball? Tape a yardstick or tape measure to a vertical surface (tree, side of house, basketball goal). While one person bounces, one or two others try to gauge the height. Try with different balls.  Figure an average of heights after 3-4 bounces.
3. Play basketball, but instead of 2 points per basket, assign certain shots specific points and keep a mental tally.
4. Get out the old Hot Wheels. Measure the distance after pushing them.  Determine ways to increase or decrease the distance. Compete with a sibling or friend to see who has the highest total after 3-4 pushes.  Depending on the age of your child, you may want to measure to the nearest foot, inch, half-inch or cm.
5. Measure the stopping distance of your bicycle.
6. Practice broad jumps in the lawn. Measure the distance you can jump. Older students can compute an average of their best 3-4 jumps. Make it a competition with siblings or friends.
7. Some good uses for a water squirt gun:
• Aim at a target with points for how close you come. The closer to the center is more points.
• Measure the distance of your squirts. What is your average distance?
• How many squirts needed to fill up a bucket?  This would be a good competition.
8. Competitive sponge race (like at school game days): Place a bucket of water at the starting line. Each player dips their sponge in and runs to the opposite side of the yard and squeezes their sponge into their own cup or bowl. Keep going back and forth. The winner is the one who fills up their container first. Find out the volume of the cup and the volume of water a sponge can hold.
9. Build a fort with scrap pieces of wood. Make a drawing to plan it. Measure the pieces to see what fits. Use glue or nails (with adult supervision).
10. Take walks around the neighborhood. Estimate the perimeter distance of the walk.
11. In the pool:
• Utilize a pool-safe squirt gun (as in #6 above).
• Estimate the height of splashes after jumping in.
• Measure the volume of the pool (l x w x h).  The result will be in cubic feet.  Convert using several online conversion calculators such as this one: https://www.metric-conversions.org/volume/
• Measure the perimeter of the pool.  If it is rectangular, does your child realize the opposite sides are equal.  This is a very important concept for students regarding geometry (opposite sides of rectangles are equal).
• What if you want to cover the pool? What would the area of the cover be?
• Measure how far you can swim.  Time the laps.  What is the average time?
12. Watch the shadows during the day. Notice the direction and the length.  Many kids don’t realize the connection between clocks and the sun. Make your own sun dial. Here are a few different resources for getting that done, some easier than others:

Indoors

1. Keep track of time needed (or allowed) for indoor activities:  30 minutes ipad, 1 hour tv, 30 minutes fixing lunch, 30 minutes for chores, etc.  This helps children get a good concept of time passage. Even many 4th and 5th graders have difficulty realizing how long a minute is.  This is also helpful as a practical application of determining elapsed time. Examples:
• It’s 11:30 now.  I’ll fix lunch in 45 minutes. What time will that be?
• I need you to be cleaned up and ready for bed at 8:30.  It’s 6:30 now.  How much time do you have?
• You can use your ipad for games for 1 hour and 20 minutes.  It is 2:30 now. What time will you need to stop?
2. Explore various recipes and practice using measuring tools.  What if the recipe calls for 3/4 cup flour and you want to double it?
3. In the bathtub, use plastic measuring cups to notice how many 1/4 cups equal a whole cup. How many 1/3 cups in a cup? How many cups in a gallon (using a gallon bucket or clean, empty milk carton)?
4. While reading, do some text analysis regarding frequency of letter usage.
• Select a passage (short paragraph).  Count the number of letters.
• Keep track of how often each letter appears in that passage. Are there letters of the alphabet never used?
• Compare with other similar length passages.
• After analyzing a few, can you make predictions about the frequency of letters in any given passage?
• How does this relate to letters requested on shows such as “Wheel of Fortune” or letters used in Scrabble?
5. Fluency in reading is a measure of several different aspects:  speed, accuracy, expression, phrasing, intonation.
• To work on the speed aspect, have your child read a selected passage (this can vary depending on the age of the child). Keep track of the time down to number of seconds. This is a baseline.
• Have the child repeat the passage to see if the time is less.  Don’t really focus on total speed because that it not helpful for a child to think good reading is super fast reading. Focus more on smoothness, accuracy and phrasing.
• Another way is to have a child read a passage and stop at 1 minute. How many words per minute were read?  Can the child increase the # of words per minute (but still keep accuracy, smoothness, and expression at a normal pace)?
6. Play Yahtzee!  Great for addition and multiplication.  Lots of other board games help with number concepts (Monopoly, etc.)
7. Lots of card games using a standard deck of cards have math links. See my last post for ideas.
8. Measure the temperature of the water in the bathtub (pool thermometers which float would be great for that). How fast does the temperature decrease. Maybe make a line graph to show the decline over time.
9. Gather up all of the coins around the house.  Read or listen to “Pigs Will be Pigs” for motivation. Keep track of how much money the pigs find around the house. Count up what was found. Use the menu in the back of the book (or use another favorite menu) to plan a meal. Be sure to check out Amy Axelrod’s other Pig books which have a math theme Amy Axelrod Pig Stories – Amazon  Here is a link to “Pigs Will be Pigs”: Pigs Will Be Pigs – Youtube version
10. Help kids plan a take-out meal that fits within the family’s budget.  Pull up Door Dash for a variety of menus or get them online from your favorite eateries. This gives great practical experience in use of the dollar to budget.
11. Look at the local weekly newspaper food advertisements.  Given a certain amount of \$, can your child pick items to help with your shopping list?  If they accompany you to the store, make use of the weighing stations in the produce section to check out the weights and cost per pound.
12. Visit your favorite online educational programs for math games or creative activities.  See a previous post regarding “Math Learning Centers.” The pattern blocks and Geoboard apps allow for a lot of creativity while experiencing the concept of “trial and error” and perseverance. These can be viewed at the website or as an app.  Here’s a link to it to save you time. Virtual math tools (cindyelkins.edublogs.org)

Please share other activities you recommend!!  Just click on the speech bubble at the top of this post or complete the comments section below.  I miss you all!

Geometry Part 7: Area and Perimeter

Today’s topic is the measurement of area and perimeter.  Even though these may be considered measurement standards, they are highly connected to geometry (such as the attributes of a rectangle). Check out previous posts in my Geometry series (Composing and Decomposing) for other mentions of area and perimeter.

Misconceptions provide a window into a child’s thinking.  If we know the misconceptions ahead of time, we can steer our teaching and directions to help students avoid them. I will go through several misconceptions and some strategies and/or lessons that might address them. Misconceptions #1-2 appear in this post. Misconceptions #3-5 will be featured in next week’s post.

Misconception #1:  A student hears this:  “We use area to measure inside a shape and perimeter to measure around a shape.”

• Problem:  The student doesn’t know how to apply this definition to real situations which require the measurement of area and/or perimeter.
• Problem:  The student may think, “Since perimeter measures the outside edge, then area means to measure the inside edge.”
• Problem:  Students confuse the two terms.

Ideas:

• Brainstorm with students (with your guidance) examples of the need for area and perimeter. Use these scenarios as you solve concrete or pictorial examples.
• Area:  garden, room size, ceiling tiles, carpet, rug, floor tiles, football field, tv screen, wallpaper, wall paint, etc.
• Perimeter:  picture frame, fencing, floor trim, wallpaper border, bulletin board border . . .
• Show this diagram which emphasizes the concept that area measures the entire inside surface using squares of different sizes (cm, inch, foot, yard, mile), while perimeter measures the rim / edge / around an object or shape usually using a ruler, tape measure, or string.
• Try this project: Use graph paper and one inch tiles (color tiles) to concretely make shapes with a given area.  Example, “Build a rectangle with an area of 24 tiles.” Specify there can be no holes in the rectangle — it must be solid. Students can experiment while moving the tiles around. Then trace the rectangle and cut it out. With this practice, students are also focusing on arrays and multiplication.  Did they find 4 ways? (2 x 12, 3 x 8, 4 x 6, 1 x 24)? Did they find out a 3 x 8 rectangle is the same as an 8 x 3 rectangle (commutative property)?
• Note:  Make sure students stay on the given lines when tracing their shapes and cutting. I found this to be difficult for many students even though I said explicitly to “Stay on the lines when you trace.”
• NO – this is not a solid rectangle. No holes allowed.

• Similar to the above:  Use the tiles and graph paper to create irregular shapes with a given area – meaning they don’t have to be rectangular. Give the directive that tiles must match at least one edge with another edge (no tip to tip accepted).  And, same as above — no holes in the shape. You can even assign different areas to each small group.  Compare shapes – put on a poster or bulletin board.
• Using the same shapes made above, determine the perimeter.  I suggest placing tick mark to keep track of what was counted. With this lesson, students will hopefully realize shapes with the same area do not necessarily have the same perimeter. Advise caution when counting corners or insets – students usually miscount these places.
• Try this project:  Design a bedroom with dimensions of (or square feet of): ___________ Include a bed, and at least one other piece of furniture or feature (nightstand, dresser, desk, shelf, chair, closet, etc.)
• The student can use smaller scale graph paper with 1 square representing 1 square foot.
• Let them have time to “work it out” and practice perseverance as a rough draft before making their final copy.
• Together, use the square foot construction paper pieces to determine an appropriate size for a bed and other furniture. Students must think of it from an overhead perspective. When I did this with a class, we settled on 3 ft. x 6 ft bed for a total of 18 square feet. We did this by laying the square foot papers on the floor to see in real life what this size looked like.
• Label the Area and Perimeter of each item in the bedroom.
• The items in the bedroom could also be made as separate cut-outs and arranged / rearranged on the “floor plan” to see all of the different ways the room could look.
• Students in 3rd grade might want to use whole units, while 4th and up might be able to use half-units.
• On a test, area answer choices would include ” ____ square inches” or “inches squared” or “inches²”  Answer choices for perimeter will omit the word “square.”

Geometry Part 5: Composing and Decomposing 3D Shapes (+ surface area)

Composing and decomposing 3D shapes should help your students become more familiar with their attributes. Here are a few activities to help.  With emphasis on hands-on methods, examining real 3D shapes may help students find edges, vertices, and faces better than pictorial models.

1.  Nets of 3D shapes are the least expensive way to get a set of 3D objects in each child’s hand, especially since most classrooms just usually have 1 set of plastic or wooden 3D shapes.
2.  Build cubes and rectangular prisms using blocks or connecting cubes.
3. Construct / deconstruct prisms using toothpicks, straws, coffee stirrers, craft sticks, or pretzel sticks as the edges. For the vertices, use clay, playdough, gum drops or slightly dried out marshmallows.
4. Lucky enough to have a set of tinker toys? Or Magna Tiles? (We got our grandson some Magna Tiles and he loves them!  These tiles have magnetic edges which can hook together in an instant. Creating a cube, rectangular prism, pyramid, etc. is easy!  They are kind of expensive, but very versatile and creative.)
5. Teach students how to draw 3D shapes. When composing a 3D shape, a student becomes more aware of the 2D faces, the edges, and the vertices they are drawing. Plus, if needed the student can draw the 3D shape on paper to assist them if taking a computer based assessment.  Here is my tutorial (below), but I’ll also include a couple of good websites in case you are 3D challenged. Click HERE for the pdf of the templates below.
6. Observe how students count the edges, vertices, and faces.  If they are randomly trying to count them, they likely will be incorrect.  When needed, show them how to be methodical with their counting (ie: When counting the edges of a cube, run your finger along the edge as you count. Count the top 4 edges, then the bottom 4 edges, then the 4 vertical edges = 12.)

One of my favorite lessons regarding decomposing shapes is when teaching students (5th grade and up) how to measure surface area.  Click HERE for the free pdf guide for creating the rectangular prisms shown below.  It includes a blank grid so you can create your own (all courtesy of http:illuminations.nctm.org using their “dynamic paper” lesson). Continue reading

Geometry Part 1: The Basics

For many schools, it seems as if Geometry and Measurement standards remain some of the lowest scored. This has always puzzled me because it’s the one area in math that is (or should be) the most hands-on — which is appealing and more motivating to students. Who doesn’t like creating with pattern blocks, making 2 and 3D shapes with various objects, using measurement tools, and getting the chance to leave your seat to explore all the classroom has to offer regarding these standards? So what is it about geometry and measurement that is stumping our students? Here are some of my thoughts – please feel free to comment and add your own:

• Vocabulary? (segment, parallel, trapezoid, perpendicular, volume, area, perimeter, etc.)
• Lack of practical experience? Not all homes have materials or provide opportunities for students to apply their knowledge (like blocks, Legos, measuring cups for cooking, tape measures for building, etc.).
• Background knowledge about the size of actual objects? We take it for granted students know a giraffe is taller than a pickup truck. But if students have not had the chance to go to a zoo, then when they are presented a picture of the two objects they might not really know which is taller / shorter. Think of all of the examples of how we also expect students to know the relative weights of objects. Without background knowledge or experience, this could impede them regarding picture type assessments.
• Standards keep getting pushed to lower grades when students may not have reached the conservation stage? If they think a tall slender container must hold more than a shorter container with a larger diameter, or they think a sphere of clay is less than the same size sphere flattened out, they may have difficulty with many of the geometry and measurement standards.

In this post, I will focus on Geometry. Here is a basic look at the geometry continuum (based on OK Stds.):

KG:  Recognize and sort basic 2D shapes (circle, square, rectangle, triangle). This includes composing larger shapes using smaller shapes (with an outline available).

1st:  Recognize, compose, and decompose 2D and 3D shapes. The new 2D shapes are hexagon and trapezoid. 3D shapes include cube, cone, cylinder, sphere.

2nd: Analyze attributes of 2D figures. Compose 2D shapes using triangles, squares, hexagons, trapezoids and rhombi. Recognize right angles and those larger or smaller than right angles.

3rd:  Sort 3D shapes based on attributes. Build 3D figures using cubes. Classify angles: acute, right, obtuse, straight.

4th:  Name, describe, classify and construct polygons and 3D figures.  New vocabulary includes points, lines, segments, rays, parallel, perpendicular, quadrilateral, parallelogram, and kite.

5th: Describe, classify, and draw representations of 2D and 3D figures. Vocabulary includes edge, face, and vertices. Specific triangles include equilateral, right, scalene, and isosceles.

Here are a couple of guides that might help you with definitions of the various 2D shapes. The 2D shapes guide is provided FREE here in a PDF courtesy of math-salamander.com.  I included a b/w version along with my colored version. The Quadrilateral flow chart I created will help you see that some shapes can have more than one name. Click on the link for a free copy (b/w and color) of the flow chart. Read below for more details about understanding the flow chart.